K-Tropicalization in Non-Archimedean Geometry

• K-tropicalization is a framework for extending tropical geometry to non-Archimedean fields, capturing valuations, Berkovich analytifications, and polyhedral structures.
• It generalizes the Kajiwara–Payne map, serving as a non-Archimedean analogue of the complex logarithmic moment map with functorial and moduli-theoretic extensions.
• The approach refines tropicalization using K-theory by incorporating Hilbert-polynomial data and Gröbner stratifications to detect scheme-theoretic nuances.

K-tropicalization denotes tropicalization in a non-Archimedean valued setting, with the toric prototype given by the Kajiwara–Payne map from a Berkovich analytification to a polyhedral target. In the toric case, for a split torus $T\simeq \mathbb G_m^n$ with character lattice $M$, cocharacter lattice $N=\operatorname{Hom}(M,\mathbb Z)$, and a $T$-toric variety $X=X(\Delta)$ associated to a rational polyhedral fan $\Delta\subset N_{\mathbb R}$, the map

$\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$

is the non-Archimedean analogue of the complex logarithmic moment map. More broadly, tropicalization over a valued field $K$ is formulated on Berkovich analytifications, can be characterized by initial degenerations, and admits functorial, logarithmic, and moduli-theoretic extensions. In more recent work, the same expression also names a $K$-theoretic refinement of tropicalization attached to logarithmic quotient sheaves, where the tropical support is decorated by Hilbert-polynomial data on every stratum rather than only by top-dimensional multiplicities (Ulirsch, 2014, Gubler, 2011, Kennedy-Hunt et al., 11 Aug 2025).

1. Terminology and scope

In one standard usage, $K$-tropicalization means tropicalization over a non-Archimedean ground field $M$0 or valued field $M$1. In the toric setting, the terminology is used because the construction is defined over a non-Archimedean ground field and serves as the non-Archimedean analogue of the complex logarithmic map. In another usage, developed in the logarithmic Quot-space literature, $M$2-tropicalization refers to a $M$3-theoretic enhancement of ordinary tropicalization that is sensitive to scheme structure and coherent sheaf data rather than only to cycle-theoretic support (Ulirsch, 2014, Kennedy-Hunt et al., 11 Aug 2025).

The literature also distinguishes between set-theoretic, topological, and scheme-theoretic forms of tropicalization. Some sources formulate the theory as tropicalization over a valued field $M$4 with valuation $M$5, rather than using the phrase $M$6-tropicalization explicitly. In that formulation, Berkovich analytification, Kajiwara–Payne tropicalization, higher-rank tropicalization, Thuillier analytification, and Ulirsch tropicalization can all be treated as realizations of a common scheme-theoretic or moduli-theoretic construction (Giansiracusa et al., 2014, Lorscheid, 2015).

2. Tropicalization over valued fields

For a valued field $M$7, a split torus $M$8, and dual lattice $M$9, the basic tropicalization map on the Berkovich analytification is

$N=\operatorname{Hom}(M,\mathbb Z)$0

For a closed subscheme $N=\operatorname{Hom}(M,\mathbb Z)$1, its tropicalization is

$N=\operatorname{Hom}(M,\mathbb Z)$2

This formulation works over arbitrary non-Archimedean valued fields, including trivially valued fields, and is compatible with base change: if $N=\operatorname{Hom}(M,\mathbb Z)$3 extends $N=\operatorname{Hom}(M,\mathbb Z)$4, then $N=\operatorname{Hom}(M,\mathbb Z)$5. The Bieri–Groves theorem identifies $N=\operatorname{Hom}(M,\mathbb Z)$6 as a finite union of $N=\operatorname{Hom}(M,\mathbb Z)$7-rational polyhedra, and if $N=\operatorname{Hom}(M,\mathbb Z)$8 is pure $N=\operatorname{Hom}(M,\mathbb Z)$9-dimensional, the polyhedra may be chosen $T$0-dimensional. The fundamental theorem takes the form

$T$1

linking tropicalization to initial degenerations (Gubler, 2011).

For toric varieties, this general valued-field construction specializes to the Kajiwara–Payne map. On an affine toric chart

$T$2

the tropicalization map is

$T$3

Equivalently, tropicalization records the valuations of torus characters at the analytic point $T$4. The target $T$5 is the partial compactification associated to the fan $T$6, and this toric model is the reference point for many later generalizations (Ulirsch, 2014).

3. Toric quotients, skeleta, and analytic stacks

A distinctive feature of the non-Archimedean toric theory is the role of the big affinoid torus

$T$7

This is an analytic subgroup of $T$8, but its underlying set does not carry a group structure in the ordinary topological sense. Consequently, the quotient $T$9 is not available as a naive topological quotient; the natural replacement is the analytic stack quotient $X=X(\Delta)$0 (Ulirsch, 2014).

The central theorem in this setting states that there is a natural homeomorphism

$X=X(\Delta)$1

such that the diagram

$X=X(\Delta)$2

commutes. Thus the Kajiwara–Payne tropicalization map is exactly the underlying topological quotient map of the analytic stack quotient by the big affinoid torus. The proof passes through a theory of non-Archimedean analytic stacks, analytic groupoids, quotient stacks, and underlying topological spaces of analytic stacks (Ulirsch, 2014).

The same paper identifies tropicalization with the non-Archimedean skeleton $X=X(\Delta)$3. There is a strong deformation retraction

$X=X(\Delta)$4

onto $X=X(\Delta)$5, together with a homeomorphism

$X=X(\Delta)$6

so tropicalization may be read equally as passage to the skeleton and as passage to the stack quotient. This is the exact non-Archimedean analogue of the complex statement that the logarithmic absolute value map is the quotient by the compact torus $X=X(\Delta)$7 and admits a section with image the locus of non-negative points (Ulirsch, 2014).

4. Logarithmic, universal, and scheme-theoretic extensions

For a fine and saturated logarithmic scheme $X=X(\Delta)$8 locally of finite type over a trivially valued field $X=X(\Delta)$9, tropicalization can be formulated as a map

$\Delta\subset N_{\mathbb R}$0

where $\Delta\subset N_{\mathbb R}$1 is Thuillier analytification and $\Delta\subset N_{\mathbb R}$2 is the canonical extension of the cone complex associated to the logarithmic structure. Locally, given a monoid chart $\Delta\subset N_{\mathbb R}$3, one has

$\Delta\subset N_{\mathbb R}$4

Globally, the construction is mediated by the characteristic morphism to a Kato fan, is functorial for morphisms of fine and saturated logarithmic schemes, and in the logarithmically smooth case recovers Thuillier’s strong deformation retraction onto the skeleton. In the toric case, $\Delta\subset N_{\mathbb R}$5, $\Delta\subset N_{\mathbb R}$6, and $\Delta\subset N_{\mathbb R}$7 agrees with the Kajiwara–Payne map (Ulirsch, 2013).

A complementary universal perspective constructs a universal embedding $\Delta\subset N_{\mathbb R}$8 into a $\Delta\subset N_{\mathbb R}$9-scheme with an $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$0-model. The tropicalization with respect to this universal embedding has underlying set canonically identified with the Berkovich analytification $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$1, and with the strong Zariski topology it is homeomorphic to the Berkovich topology. In the affine case, the universal tropicalization is the inverse limit of the tropicalizations over all affine embeddings, refining Payne’s inverse-limit theorem from topological spaces to scheme-theoretic tropicalizations. The same construction represents a moduli functor of semivaluations compatible with the base valuation (Giansiracusa et al., 2014).

A further unifying formulation uses ordered blueprints. In that framework, tropicalization is a representing object for the functor of extensions of a valuation $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$2. If $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$3 is totally positive, tropicalization becomes a base change along $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$4; if $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$5 is idempotent, the bend relation yields a scheme-theoretic tropicalization. Berkovich analytification and Kajiwara–Payne tropicalization appear as $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$6-rational point sets of this scheme-theoretic object, and the same formalism also recovers higher-rank tropicalization, Giansiracusa tropicalization, Macpherson analytification, Thuillier analytification, and Ulirsch tropicalization (Lorscheid, 2015).

5. Topological and combinatorial structure

A fundamental structural theorem states that if $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$7 is irreducible and $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$8 is either algebraically closed, complete, or real closed with convex valuation ring, then $\trop_\Delta:X^{an}\longrightarrow N_{\mathbb R}(\Delta)$9 is connected through codimension $K$0. Here a pure-dimensional polyhedral complex is connected through codimension $K$1 if any two facets can be joined by a sequence of facets in which consecutive facets meet along a codimension-$K$2 face. The theorem applies in particular to trivial valuation, because the trivial valuation is complete, and it answers affirmatively a question posed by Einsiedler, Lind, and Thomas (Cartwright et al., 2012).

In characteristic $K$3, the connectivity statement strengthens. If $K$4 is an irreducible $K$5-dimensional variety, $K$6 is a pure $K$7-dimensional rational polyhedral complex with $K$8, and $K$9 is the dimension of the lineality space, then $K$0 is $K$1-connected through codimension one. The same work proves a tropical Bertini theorem: for a generic rational affine hyperplane $K$2, the slice $K$3 is again the tropicalization of an irreducible variety. This gives a realizability obstruction: a pure polyhedral complex that is not sufficiently connected through codimension one cannot be the tropicalization of an irreducible variety in characteristic $K$4 (Maclagan et al., 2019).

The valued-field framework also extends beyond rank-one valuations. Over a higher-dimensional local field $K$5, the valuation takes values in a lexicographically ordered group $K$6, and tropicalization lives in

$K$7

For a $K$8-dimensional irreducible closed subscheme of a torus, the tropicalization is a rational polyhedral complex of dimension $K$9. In this higher-rank setting, hypersurface tropicalization is still described as the non-differentiability locus of a tropical polynomial, but the geometry becomes layered by the rank-$K$0 valuation (Banerjee, 2011).

A different refinement occurs over real closed non-Archimedean valued fields. Real tropicalization remembers sign as well as valuation: $K$1 For a semialgebraic set $K$2, the associated real analytification $K$3 is homeomorphic to the inverse limit of all real tropicalizations of $K$4. The real tropical fundamental theorem describes the tropicalization of $K$5 by finitely many tropical inequalities, orthant by orthant, and shows that the sign pattern cannot in general be ignored (Jell et al., 2018).

6. Moduli-theoretic realizations

Tropicalization interacts especially strongly with moduli spaces. For the Deligne–Mumford–Knudsen moduli stack $K$6, the skeleton is naturally identified with the moduli space of extended tropical curves: $K$7 The naive tropicalization map on $K$8, defined by taking edge lengths to be valuations of smoothing parameters $K$9 appearing in local node equations $M$00, agrees with the projection to the skeleton. Under this identification, the tropicalization map is continuous, proper, and surjective, and the tropical forgetful, clutching, and gluing maps are compatible with their algebraic counterparts (Abramovich et al., 2012).

For stable maps into a $M$01-analytic space $M$02 equipped with a strictly semi-stable formal model $M$03, the tropical target is the Clemens polytope $M$04. Tropicalization sends an analytic stable map to a parametrized tropical curve in $M$05; its edges carry weight vectors extracted from annuli in the source curve, and these weights satisfy a balancing condition expressed cohomologically through nearby cycles, vanishing cycles, and intersection numbers on strata of the formal model. The moduli space $M$06 of simple parametrized tropical curves of bounded tropical degree is compact and stratified by open convex polyhedra, and the tropicalization map from the analytic moduli stack of stable maps is continuous, with compact polyhedral image (YU, 2014).

7. The $M$07-theoretic refinement

In the logarithmic Quot-space setting, a logarithmic quotient sheaf

$M$08

on an expansion $M$09 defines a Gröbner stratification

$M$10

by identifying points with the same initial degeneration. For each cell $M$11 of the expansion, the associated $M$12-weight is the multivariable Hilbert polynomial

$M$13

where $M$14. The $M$15-tropicalization is the Gröbner stratification together with these $M$16-weights. Unlike ordinary tropicalization, which is tied to Chow theory and top-dimensional multiplicities, this construction records coherent-sheaf data on every stratum, can detect embedded and nonreduced structure, and allows negative contributions because the decorations are Euler characteristics rather than positive multiplicities (Kennedy-Hunt et al., 11 Aug 2025).

This refinement is controlled by a canonical transversalization theorem. For a coherent sheaf $M$17 on an snc pair $M$18, there is a canonical piecewise-linear stratification $M$19 such that a smooth subdivision $M$20 makes the strict transform algebraically transverse if and only if $M$21 refines $M$22. Equivalently, there is a canonical logarithmic space $M$23 through which every snc logarithmic blowup making $M$24 algebraically transverse factors (Kennedy-Hunt et al., 11 Aug 2025).

The balancing condition for these $M$25-decorated tropicalizations is derived from the $M$26-theory of toric bundles. Using the presentation of the Grothendieck ring due to Sankaran–Uma, the theory imposes relations coming from toric divisors and their regular crossings, and these relations are the $M$27-theoretic analogue of ordinary tropical balancing. One consequence is a strong finiteness theorem: for fixed numerical data, the set of combinatorial types of $M$28-tropicalizations is finite, and $M$29-tropicalizations with fixed numerics are parametrized by a finite-dimensional polyhedral complex. This combinatorial boundedness is the key input in proving boundedness and properness of logarithmic Quot spaces and logarithmic Hilbert spaces (Kennedy-Hunt et al., 11 Aug 2025).

In this refined sense, $M$30-tropicalization is not merely tropicalization over a field $M$31; it is a sheaf-theoretic tropical invariant whose relationship to $M$32-theory parallels the relationship of ordinary tropicalization to Chow theory. The resulting framework places Gröbner stratifications, state polytopes, secondary polytopes, and logarithmic moduli problems inside a single polyhedral and $M$33-theoretic formalism (Kennedy-Hunt et al., 11 Aug 2025).